Vacuum measurement: McLeod, Pirani and ionisation gauges

Why it matters

Vacuum is used in freeze drying, distillation of heat-sensitive products, vacuum furnaces, thin-film coating, electron microscopes and semiconductor fabrication. Below a few hundred pascals, elastic gauges and manometers lose sensitivity, so plants rely on gauges that measure a property of the rarefied gas: its compressibility (McLeod), its thermal conductivity (Pirani, thermocouple gauge) or how many ions it produces (ionisation gauges). Choosing the right gauge for the range, and knowing which ones depend on the gas composition, is a routine design decision.

Key ideas

Vacuum ranges (approximate). Rough vacuum: atmospheric down to about 100 Pa; medium vacuum: 100 Pa to 0.1 Pa; high vacuum: 0.1 Pa to about 10⁻⁵ Pa; ultra-high vacuum: below about 10⁻⁵ Pa. 1 Torr ≈ 1 mm Hg ≈ 133.3 Pa; 1 mbar = 100 Pa.

McLeod gauge (compression gauge).

• A known volume V of gas at the unknown pressure p is trapped in a glass bulb by raising mercury, then compressed into a sealed capillary of bore area a. Boyle's law (isothermal) gives the original pressure from the compressed one.
• Square-law method: mercury in the open reference capillary is raised until it is level with the top of the sealed capillary. The trapped gas column then has length h, and its pressure exceeds p by ρgh. This gives p ≈ ρ·g·a·h²/V, so the scale is non-linear (square law).
• It is an absolute gauge: the calibration follows from dimensions only, independent of gas type, so it is used to calibrate Pirani and ionisation gauges.
• Limitations: intermittent (not continuous) reading, mercury handling, and it does not read condensable vapours correctly, because vapours condense when compressed and do not obey Boyle's law. Typical range from about 10 Pa down to about 10⁻³ Pa, depending on bulb and capillary size.

Thermal-conductivity gauges (Pirani and thermocouple).

• A heated filament loses heat by conduction through the gas, by radiation and through its supports. At low pressure (when the molecular mean free path is comparable to the filament-to-wall gap) the gas conduction is roughly proportional to pressure.
• Pirani: the filament (platinum or tungsten) forms one arm of a Wheatstone bridge. As pressure falls, less heat is conducted away, the filament runs hotter and its resistance rises, unbalancing the bridge. In the constant-temperature version the bridge voltage is servoed to keep the filament resistance fixed, and the heating power is the measure of pressure. A sealed, evacuated compensating filament in the adjacent arm cancels ambient temperature changes.
• Thermocouple gauge: constant heater current; a thermocouple welded to the heater reads its temperature, which rises as pressure falls.
• Useful range roughly 0.1 Pa to 1 kPa. Above this the gas conductivity becomes nearly independent of pressure; below it radiation and end losses swamp the gas conduction.
• The reading depends on the gas, because thermal conductivity differs (hydrogen and helium conduct far better than air). Readings for other gases need correction factors or calibration.

Ionisation gauges.

• Hot-cathode (triode or Bayard–Alpert): electrons from a heated filament are accelerated towards a positive grid; they ionise gas molecules, and the positive ions are collected by a negative collector. The ion current is proportional to the molecular density: Ii = S·Ie·p. The Bayard–Alpert design uses a fine wire collector inside the grid to reduce the X-ray limit and reaches about 10⁻⁸ Pa or lower.
• Cold-cathode (Penning): a high voltage discharge in a magnetic field; no filament to burn out, but less accurate.
• Range about 10⁻¹ Pa downwards; above that the filament can burn out and the current is no longer linear. The sensitivity S depends on gas type (ionisation cross-section).

Choosing a gauge. Systems usually carry two gauges: a Pirani for rough/medium vacuum and an ionisation gauge switched on only after the Pirani shows the pressure is low enough. A McLeod (or a capacitance diaphragm gauge, which is gas-independent) serves as the reference.

Formulas

p·V = (p + ρ·g·h)·a·h (McLeod, square-law, Boyle's law on the trapped gas)

p = ρ·g·a·h² / (V − a·h) ≈ ρ·g·a·h² / V (McLeod, when a·h ≪ V)

Ii = S·Ie·p (hot-cathode ionisation gauge)

P_el = P₀ + C·p (Pirani in its linear low-pressure region; idealised)

Symbols: p = unknown pressure (Pa); V = volume trapped (bulb plus capillary) (m³); a = bore area of the sealed capillary (m²); h = length of compressed gas column = difference in mercury levels (m); ρ = density of mercury (13 600 kg/m³); g = 9.81 m/s²; Ii = ion current (A); Ie = electron emission current (A); S = gauge sensitivity (Pa⁻¹), gas-dependent; P_el = electrical power to hold the filament at constant temperature (W); P₀ = radiation and end losses (W); C = constant from calibration (W/Pa).

Worked examples

Example 1 (standard): ionisation gauge. Given: Bayard–Alpert gauge, sensitivity for nitrogen S = 0.075 Pa⁻¹ (about 10 Torr⁻¹), emission current Ie = 4 mA, ion current Ii = 3 nA.

1. p = Ii / (S·Ie).
2. S·Ie = 0.075 × 4 × 10⁻³ = 3.0 × 10⁻⁴ A/Pa.
3. p = 3 × 10⁻⁹ / 3.0 × 10⁻⁴ = 1.0 × 10⁻⁵ Pa.
4. Answer: p = 1.0 × 10⁻⁵ Pa (high vacuum). For a different gas, S changes and the reading must be corrected.

Example 2 (GATE level): McLeod gauge. Given: bulb plus capillary volume V = 100 cm³, sealed capillary bore 1.0 mm, square-law reading h = 20 mm of mercury, ρ = 13 600 kg/m³.

1. a = π·d²/4 = π × (1.0 × 10⁻³)² / 4 = 7.854 × 10⁻⁷ m².
2. a·h = 7.854 × 10⁻⁷ × 0.020 = 1.571 × 10⁻⁸ m³, which is negligible against V = 1.0 × 10⁻⁴ m³.
3. p ≈ ρ·g·a·h² / V = 13 600 × 9.81 × 7.854 × 10⁻⁷ × (0.020)² / 1.0 × 10⁻⁴.
4. ρ·g·h = 13 600 × 9.81 × 0.020 = 2668.3 Pa; compression ratio a·h/V = 1.571 × 10⁻⁴.
5. p = 2668.3 × 1.571 × 10⁻⁴ = 0.419 Pa.
6. Answer: p ≈ 0.42 Pa. Note the gas was compressed about 6400 times, which is why the gauge can read such low pressures with a mercury column.

Example 3 (Pirani, idealised linear region). A constant-temperature Pirani needs 10.0 mW at very high vacuum (P₀) and 14.0 mW at a calibration pressure of 1.0 Pa. Then C = (14.0 − 10.0)/1.0 = 4.0 mW/Pa. At 12.0 mW: p = (12.0 − 10.0)/4.0 = 0.50 Pa. Real Pirani curves are only linear near the low end, so manufacturers supply full calibration curves.

Common mistakes

• Using the McLeod gauge on water vapour or solvent vapour: the vapour condenses on compression and the gauge reads low.
• Forgetting that the McLeod square-law scale is non-linear (p ∝ h²).
• Applying an air-calibrated Pirani or ionisation reading directly to helium, hydrogen or argon without the gas correction factor.
• Switching on a hot-cathode ionisation gauge at rough vacuum; the filament oxidises or burns out.
• Mixing Torr, mbar and Pa: 1 Torr = 133.3 Pa, 1 mbar = 100 Pa.
• Thinking a Pirani wire cools as vacuum improves; it gets hotter because less heat is conducted away.

For GATE IN

Questions are mostly conceptual: which gauge suits which range, which are absolute and gas-independent (McLeod, capacitance diaphragm), which property each senses, and why a bridge with a compensating element is used in the Pirani gauge. Numericals use Boyle's law in the McLeod gauge (often the square-law relation) and the ion-current relation. Practise unit conversion between Pa, mbar and Torr.

Quick check

1. Which vacuum gauge is absolute and used to calibrate others?
2. Why does a Pirani gauge lose sensitivity above about 1 kPa?
3. In a McLeod gauge, how does p scale with the reading h in the square-law method?
4. An ionisation gauge has S = 0.1 Pa⁻¹ and Ie = 2 mA. What pressure gives Ii = 2 nA?
5. Why is the McLeod gauge unsuitable for water vapour?

Answers: 1. McLeod gauge. 2. Gas conductivity becomes nearly independent of pressure. 3. p ∝ h². 4. 1 × 10⁻⁵ Pa. 5. The vapour condenses on compression and does not follow Boyle's law.